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PSLE

PSLE Foundation Mathematics

<p>Express $y=2x^2-12x+11$ in completed-square form and state its minimum value.</p>

Express $y=2x^2-12x+11$ in completed-square form and state its minimum value.

Completing the Square and Vertex Form

Maximum and Minimum Values of General Quadratic Functions

Minimum value by completing the square

Quadratic Functions, Equations and Inequalities

<p>The equation of a curve is $y = -2x^2 + 3x + 5$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Write $y = -2x^2 + 3x + 5$ in the form $y = a(x - h)^2 + k$, where&nbsp;$a$, $h$ and $k$ are constants.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Using your results in part <strong>(a)</strong>, explain clearly why the maximum value of $y$ is $k$ when&nbsp;$x = h$.<span class="inline-marks">[2]</span></div></li></ol>

The equation of a curve is $y = -2x^2 + 3x + 5$.

2024 GESS PRELIMS P1 Q1

<ol class="i-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Express $4x^2 + 8x - 5$ in the form $p(x + q)^2 + r$, where&nbsp;$p$, $q$ and $r$ are constants to be found.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Hence, state the coordinates of the turning point of the curve $y = 4x^2 + 8x - 5$.<span class="inline-marks">[1]</span></div></li></ol>

2024 FMSS PRELIMS P1 Q1

<p>The profit, $\$y$ of a company can be modelled by the equation $y = a(x - h)^2 + k$, where&nbsp;$x$ is the number of goods sold and $a$, $h$ and $k$ are constants.<br>The company obtained the maximum profit of $\$24\,500$ when&nbsp;$800$ goods were sold.<br>The company incurred a loss of $\$7\,500$ when no goods were sold.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">State the value of $h$ and of $k$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph"><p>Using the values of $h$ and $k$ found in part <strong>(a)</strong>, find the value of $a$.</p><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Find the range of the number of goods the company needed to sell to earn a profit.<span class="inline-marks">[2]</span></div></li></ol>

The profit, $\$y$ of a company can be modelled by the equation $y = a(x - h)^2 + k$, where $x$ is the number of goods sold and $a$, $h$ and $k$ are constants.
The company obtained the maximum profit of $\$24\,500$ when $800$ goods were sold.
The company incurred a loss of $\$7\,500$ when no goods were sold.

Quadratic Inequalities

Applications of Quadratic Functions and Equations

2024 CGSS P1 Q2

<p>The cross-section of a skateboard half-pipe is modelled by the quadratic function $y =  \frac{1}{2}x^2 - 4x + 9$, where&nbsp;$x$ metres represents the horizontal distance from left end of the half-pipe, and $y$ metres represents the height above the ground.</p>
<p>The graph of the function models the curved surface of the half-pipe.</p>
<ol class="i-list"><li data-list-item-id="item-1779863970301-8i7w7kccu" data-list-item-path="item-1779863970301-8i7w7kccu"><div class="list-item-paragraph"><p>Show that the equation of the curve can be written in the form $y = \frac{1}{2}(x - 4)^2 + 1.$</p><p>Hence, state the coordinates of the turning point.</p></div></li><li data-list-item-id="item-1779864013798-i6efiat6a" data-list-item-path="item-1779864013798-i6efiat6a"><div class="list-item-paragraph">Explain what the turning point represents in the context of the half-pipe.</div></li><li data-list-item-id="item-1779864053174-je0mqtrit" data-list-item-path="item-1779864053174-je0mqtrit"><div class="list-item-paragraph">Sketch the graph of the quadratic function clearly showing:</div><ul class="bullet-list"><li data-list-item-id="subitem-1779864068279-6voek6d7l" data-list-item-path="item-1779864053174-je0mqtrit/subitem-1779864068279-6voek6d7l"><div class="list-item-paragraph">the turning point,</div></li><li data-list-item-id="subitem-1779864074574-hmuf5w09s" data-list-item-path="item-1779864053174-je0mqtrit/subitem-1779864074574-hmuf5w09s"><div class="list-item-paragraph"><p>the intercept with the $y$-axis,</p></div></li><li data-list-item-id="subitem-1779864074994-t5ucj8kvb" data-list-item-path="item-1779864053174-je0mqtrit/subitem-1779864074994-t5ucj8kvb"><div class="list-item-paragraph">the axis of symmetry.</div></li></ul></li><li data-list-item-id="item-1779864093967-t3qlykxyu" data-list-item-path="item-1779864093967-t3qlykxyu"><div class="list-item-paragraph">If the right end of the half pipe has a height of $13.5$ m from the ground, find the width of the half pipe from one end to the other.</div></li></ol>

The cross-section of a skateboard half-pipe is modelled by the quadratic function $y =  \frac{1}{2}x^2 - 4x + 9$, where $x$ metres represents the horizontal distance from left end of the half-pipe, and $y$ metres represents the height above the ground.

The graph of the function models the curved surface of the half-pipe.

Sketching Quadratic Graphs

<p>An object was launched from a machine. Its height, $y$ metres above ground, $t$ seconds after launch, can be modelled by a quadratic function $y = f(t)$. Four seconds after launch, the object reaches its maximum height of $81$ m and then dropped to a height of $54$ m after a further $3$ seconds. </p>
<ol class="abc-list"><li data-list-item-id="item-1773454586595-nxo7pguih"><div class="list-item-paragraph">Find the initial height of the object when it was launched.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-1773454587503-2ajg6lta7"><div class="list-item-paragraph">Without solving a quadratic equation, explain and determine how long did the object stay above $54$ m ?<span class="inline-marks">[2]</span></div></li></ol>

An object was launched from a machine. Its height, $y$ metres above ground, $t$ seconds after launch, can be modelled by a quadratic function $y=f(t)$. Four seconds after launch, the object reached its maximum height of 81 m and then dropped to a height of 54 m after a further three seconds.

Find the initial height of the object when it was launched.

Without solving a quadratic equation, explain and determine how long did the object stay above 54 m?

2025 CGSS PRELIMS P2 Q1 

<p>Given the curve $y = kx^2 + 2kx - 3$, where&nbsp;$k$ is a constant.</p>
<ol class="abc-list"><li data-list-item-id="item-a-9"><div class="list-item-content"><span class="list-item-text">By expressing $y$ in the form $k(x+b)^2 - 1$, determine the value of $k$ and of $b$.</span><span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b-9"><div class="list-item-content"><span class="list-item-text">Hence, show that the curve is always below the $x$-axis.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1772957050027-bldwukdfi"><div class="list-item-paragraph">State the turning point of the curve and determine the nature of this point.<span class="inline-marks">[2]</span></div></li></ol>

Given the curve $y=kx^2+2kx-3$, where $k$ is a constant.

By expressing $y$ in the form $k(x+b)^2-1$, determine the value of $k$ and of $b$.

Hence, show that the curve is always below the $x$-axis.

State the turning point of the curve and determine the nature of this point.

Quadratic Expressions Positive or Negative for All Real Values

2025 AHS PRELIMS P1 Q5

<ol class="abc-list"><li data-list-item-id="item-a-2" data-list-item-path="item-a-2"><div class="list-item-paragraph">A ball is thrown vertically upwards. Its height $h$ m, above the ground at time $t$ seconds after being thrown is given by the formula $h = 7 + 20t - 5t^2$.<br>Find the maximum height attained by the ball and the time at which this occurs.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b-2" data-list-item-path="item-b-2"><div class="list-item-content"><span class="list-item-text">Find the set of values of the constant $k$ for which the curve $y = x^2 + 6x + k$ lies entirely above the line $y = k(x - 1)$.</span><span class="inline-marks">[4]</span></div></li></ol>

A ball is thrown vertically upwards. Its height $h$ m, above the ground at time $t$ seconds after being thrown is given by the formula $h=7+20t-5t^2$. Find the maximum height attained by the ball and the time at which this occurs.

Find the set of values of the constant $k$ for which the curve $y=x^2+6x+k$ lies entirely above the line $y=k(x-1)$.

2025 DSS PRELIMS P2 Q1

<p>Explain why $3x^2 + 15x + 20$ cannot be smaller than $1$.<span class="inline-marks">[3]</span></p>

Explain why $3x^2+15x+20$ cannot be smaller than 1.

2025 DSS PRELIMS P1 Q3

<p>The diagram shows the graph of $y = x^2 - x - 5$ for&nbsp;$-3 \le x \le 4$.<br>Use the graph to answer the following questions. Present your solutions on the next page.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769516562543-yh7ojfj8sr.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk1MTY1NjI1NDMteWg3b2pmajhzci5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODY3OTE2LCJleHAiOjE3OTI4NjgyMTZ9.r39hu4MtVzbDJA8Vnym3clPS0ojj_fXNgLoSwAwmBUc" style="max-width: min(600px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph"><strong>Using the above graph,</strong></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">find the value of $y$ when&nbsp;$x = 3$</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">write down the coordinates of the turning point,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">write down the equation of the line of symmetry,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">estimate the solutions of the equation $x^2 - x - 5 = 0$.</span><span class="inline-marks">[1]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">A suitable straight line is to be drawn on the same diagram to solve the equation $x^2 - 3x - 2 = 0$.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">State the equation of the straight line to be drawn.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain how the straight line in <strong>(ii)(a)</strong> drawn on the graph can lead to an estimation of the solution(s) to the equation $x^2 - 3x - 2 = 0$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve $x^2 - 3x - 2 = 0$ graphically.</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

The diagram shows the graph of $y = x^2 - x - 5$ for $-3 \le x \le 4$.
Use the graph to answer the following questions. Present your solutions on the next page.

A suitable straight line is to be drawn on the same diagram to solve the equation $x^2 - 3x - 2 = 0$.

Solving Quadratic Equations Graphically

Interpreting Quadratic Graphs

IP2 Class Test (May 2025) Q10 (2024 RGS Y2 CBA1 Q5)

<p>Express $x^2 - 2x - 1$ in the form $(x - a)^2 - b$, where $a$ and $b$ are integers. Hence,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the minimum value of $x^2 - 2x - 1$ and the value of $x$ at which the minimum value occurs.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = x^2 - 2x - 1$, stating clearly the turning point, $x$-intercepts and the $y$-intercept.</span></div></li></ol>

Express $x^2 - 2x - 1$ in the form $(x - a)^2 - b$, where $a$ and $b$ are integers. Hence,

Find the minimum value of $x^2 - 2x - 1$ and the value of $x$ at which the minimum value occurs.

Sketch the graph of $y = x^2 - 2x - 1$, stating clearly the turning point, $x$-intercepts and the $y$-intercept.

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express $4x^2 + 4x - 15$ in the form of $a(x + b)^2 + c$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, sketch the graph of $y = 4x^2 + 4x - 15$, indicating clearly the coordinates of the turning point and $y$-intercept.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain why another graph $y = -\left(x + \frac{1}{2}\right)^2 - 19$ will not intersect the graph of $y = 4x^2 + 4x - 15$.</span><span class="inline-marks">[2]</span></div></li></ol>

Express $4x^2 + 4x - 15$ in the form of $a(x + b)^2 + c$.

Hence, sketch the graph of $y = 4x^2 + 4x - 15$, indicating clearly the coordinates of the turning point and $y$-intercept.

Explain why another graph $y = -\left(x + \frac{1}{2}\right)^2 - 19$ will not intersect the graph of $y = 4x^2 + 4x - 15$.

2023 CSS AM S3 EOY Q5

<p>A golfer hits a ball such that the height, $h$ metres, of the ball above the ground $t$ seconds later is given by $h = 0.2t(40 - t)$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">By expressing the function in the form $h = a(t - m)^2 + n$, where $a$, $m$ and $n$ are constants, explain whether the ball can reach a height of $100$ metres.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the range of values of $t$ for which the ball is at most $75$ metres above the ground.</span><span class="inline-marks">[4]</span></div></li></ol>

A golfer hits a ball such that the height, $h$ metres, of the ball above the ground $t$ seconds later is given by $h = 0.2t(40 - t)$.

By expressing the function in the form $h = a(t - m)^2 + n$, where $a$, $m$ and $n$ are constants, explain whether the ball can reach a height of $100$ metres.

Find the range of values of $t$ for which the ball is at most $75$ metres above the ground.

2023 Ahmad Ibrahim Sec AM S3 EOY Q4

<p>An arched underpass has the shape of a parabola as shown. In the diagram, the curved underpass can be modelled by the quadratic function $y = a(x - h)^2 + k$, where $a$, $h$ and $k$ are constants. In this model, $x$ m is the horizontal distance from $O$ and $y$ m is the height of the arch.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768287539506-lozwia7s2nm.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgyODc1Mzk1MDYtbG96d2lhN3Mybm0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5Mjg2NzkxNiwiZXhwIjoxNzkyODY4MjE2fQ.fg8Pnjafz1zq1-qxV_QkUL2c0Dhlu6pn-FqHYt2HOPE" style="max-width: min(220px, 100%); height: auto; display: block;" alt="Diagram of an arched underpass with a vehicle" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">A one-way road passing through the arch is $4.2$ m wide and the maximum height of the arch is $4.2$ m. State the coordinates of the maximum point of the quadratic function.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the exact value of $a$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Determine whether it is possible for a vehicle that is $2.8$ m wide and $2.4$ m tall to navigate through the underpass. Explain your reasoning by showing your workings clearly.</span><span class="inline-marks">[3]</span></div></li></ol>

An arched underpass has the shape of a parabola as shown. In the diagram, the curved underpass can be modelled by the quadratic function $y = a(x - h)^2 + k$, where $a$, $h$ and $k$ are constants. In this model, $x$ m is the horizontal distance from $O$ and $y$ m is the height of the arch.

A one-way road passing through the arch is $4.2$ m wide and the maximum height of the arch is $4.2$ m. State the coordinates of the maximum point of the quadratic function.

Determine whether it is possible for a vehicle that is $2.8$ m wide and $2.4$ m tall to navigate through the underpass. Explain your reasoning by showing your workings clearly.

2023 ACSBR AM S3 EOY Q4 

<p>Sketch the graph of $y={{x}^{2}}-3x-10$, indicate clearly the turning point and cutting point(s) on the axes.</p>
<p>Hence, find</p>
<ol class="i-list"><li><div class="list-item-paragraph">the maximum or minimum point of the function and the value of $x$ where it occurs, </div></li><li><div class="list-item-paragraph">the equation of the line of symmetry of the graph.</div></li></ol>

Sketch the graph of $y={{x}^{2}}-3x-10$, indicate clearly the turning point and cutting point(s) on the axes.

<p>Express $y=-3{{x}^{2}}+6x-11$ in the form $y=a{{(x+h)}^{2}}+k$ where&nbsp;$a$, $h$ and $k$ are constants. State the maximum value of $y$ and the corresponding value of $x$.  Hence sketch the graph of $y=-3{{x}^{2}}+6x-11$ and state the coordinates of the maximum point of the curve.</p>

Express $y=-3{{x}^{2}}+6x-11$ in the form $y=a{{(x+h)}^{2}}+k$ where $a$, $h$ and $k$ are constants. State the maximum value of $y$ and the corresponding value of $x$.  Hence sketch the graph of $y=-3{{x}^{2}}+6x-11$ and state the coordinates of the maximum point of the curve.

<p>A tank is firing rounds during a target practice.  The height, $h$ metres, of a projectile above the ground during flight is governed by the equation $h=5t-\frac{1}{5}{{t}^{2}}$ where $t$ is the time in seconds for the projectile in flight.</p>
<ol class="i-list"><li><div class="list-item-paragraph">What is the maximum height achieved by a projectile?</div></li><li><div class="list-item-paragraph">Sketch the curve $h=5t-\frac{1}{5}{{t}^{2}}$ for $h\ge 0$. </div></li><li><div class="list-item-paragraph">Find the range of values of $t$ for which a round is at least $10$ metres above the ground.</div></li></ol>

A tank is firing rounds during a target practice.  The height, $h$ metres, of a projectile above the ground during flight is governed by the equation $h=5t-\frac{1}{5}{{t}^{2}}$ where $t$ is the time in seconds for the projectile in flight.

<p>Express $y=2{{x}^{2}}+4x-6$ in the form $a{{(x+b)}^{2}}+k$, where $a$, $b$ and $k$ are constants.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">State the minimum value of the function and the value of $x$ when this occurs.</div></li><li><div class="list-item-paragraph">Hence, sketch the graph of $y=2{{x}^{2}}+4x-6$ and state the coordinates of $x$ and $y$-intercepts and turning points.</div></li></ol>

Express $y=2{{x}^{2}}+4x-6$ in the form $a{{(x+b)}^{2}}+k$, where $a$, $b$ and $k$ are constants.

<p>A piece of wire, $100$ cm in length is divided into two parts.  One part is bent to form a square of side $x$ cm and the other part a square of side $y$ cm.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express $y$ in terms of $x$.</div></li><li><div class="list-item-paragraph">Show that the total area, $A$ cm$^{2}$, of the two squares is given by $A={{x}^{2}}+{{(25-x)}^{2}}$.</div></li><li><div class="list-item-paragraph">Calculate the value of $x$ for which $A$ has a stationary value. Determine whether this value of $x$ makes $A$ a maximum or a minimum.  Hence, find the value of $A$.</div></li></ol>

A piece of wire, $100$ cm in length is divided into two parts.  One part is bent to form a square of side $x$ cm and the other part a square of side $y$ cm.

Application to Real Life Problems

<p>Express $y={{x}^{2}}-8x+17$ in the form of $y={{(x-h)}^{2}}+k$. Hence, state</p>
<ol class="i-list"><li><div class="list-item-paragraph">the maximum or minimum point of the function and the value of $x$ where it occurs </div></li><li><div class="list-item-paragraph">the equation of the line of symmetry of the graph. </div></li></ol>
<p>Sketch the graph and indicate clearly the turning point and cutting point(s) on the axes.</p>

Express $y={{x}^{2}}-8x+17$ in the form of $y={{(x-h)}^{2}}+k$. Hence, state

Sketch the graph and indicate clearly the turning point and cutting point(s) on the axes.

Math Question Bank